This text is a first introduction to homological algebra, assuming only very basic prerequisites. For instance we do recall in some detail basic definitions and constructions in the theory of abelian groups and modules, though of course a prior familiarity with these ingredients will be helpful. Also we use very little category theory, if it all. Where universal constructions do appear we spell them out explicitly in components and just mention their category-theoretic names for those readers who want to dig deeper. We do however freely use the words functor and commuting diagram. The reader unfamiliar with these elementary notions should click on these keywords and follow the hyperlink to the explanation right now.

We do use some basic category theorylanguage in the following, but no actual category theory. The reader should know what a category is, what a functor is and what a commuting diagram is. These concepts are more elementary than any genuine concept in homological algebra to appear below and of general use. Where we do encounter universal constructions below we call them by their category-theoretic name but always spell them out in components explicity.

1) Homotopy type of topological spaces

This section reviews some basic notions in topology and homotopy theory. These will all serve as blueprints for corresponding notions in homological algebra.

Definition

A topological space is a setXX equipped with a set of subsetsU⊂XU \subset X, called open sets, which are closed under

Remark

In words this says that a homotopy between two continuous functions ff and gg is a continuous 1-parameter deformation of ff to gg. That deformation parameter is the canonical coordinate along the interval [0,1][0,1], hence along the “length” of the cylinderX×Δ1X \times \Delta^1.

Proposition

Concatenation of loops respects based homotopy classes where it becomes an associative, unital binary pairing with inverses, hence the product in a group.

Definition

For XX a topological space and x∈Xx \in X a point, the set of based homotopy equivalence classes of based loops in XX equipped with the group structure from prop. 3 is the fundamental group or first homotopy group of (X,x)(X,x), denoted

Definition

Remark

Equivalently this means that XX and YY have the same (weak) homotopy type if there exists a zigzag of weak homotopy equivalences

X←→←…→Y.
X \leftarrow \to\leftarrow \dots \to Y
\,.

One can understand the homotopy type of a topological space just in terms of its homotopy groups and how they act on each other. (This data is called a Postnikov tower of XX.) But computing and handling homotopy groups is in general hard, famously so already for the seemingly simple case of the homotopy groups of spheres. Therefore we now want to simplify the situation by passing to a “linear/abelian approximation”.

It is straightforward to check by explicit inspection that the evident injection and restriction maps between the sets of singular simplices make (SingX)•(Sing X)_\bullet into a simplicial set. We now briefly indicate a systematic way to see this using basic category theory, but the reader already satisfied with this statement should jump ahead to the abelianization of (SingX)n(Sing X)_n in prop. 6 below.

Remark

This is called the “simplex category” because we are to think of the object [n][n] as being the “spine” of the nn-simplex. For instance for n=2n = 2 we think of 0→1→20 \to 1 \to 2 as the “spine” of the triangle. This becomes clear if we don’t just draw the morphisms that generate the category [n][n], but draw also all their composites. For instance for n=2n = 2 we have_

Remark (aside)

It turns out that that homotopy type of the topological space XX is entirely captured by its singular simplicial complex SingXSing X (this is the content of the homotopy hypothesis-theorem).

Now we abelianize the singular simplicial complex (SingX)•(Sing X)_\bullet in order to make it simpler and hence more tractable.

Definition

A formal linear combination of elements of a setS∈S \in Set is a function

a:S→ℤ
a : S \to \mathbb{Z}

such that only finitely many of the values as∈ℤa_s \in \mathbb{Z} are non-zero.

Identifying an element s∈Ss \in S with the function S→ℤS \to \mathbb{Z}, which sends ss to 1∈ℤ1 \in \mathbb{Z} and all other elements to 0, this is written as

a=∑s∈Sas⋅s.
a = \sum_{s \in S} a_s \cdot s
\,.

In this expression one calls as∈ℤa_s \in \mathbb{Z} the coefficient of ss in the formal linear combination.

Remark

For S∈S \in Set, the group of formal linear combinationsℤ[S]\mathbb{Z}[S] is the group whose underlying set is that of formal linear combinations, def. 19, and whose group operation is the pointwise addition in ℤ\mathbb{Z}:

Definition

A chain complex of abelian groupsC•C_\bullet is a collection {Cn∈Ab}n\{C_n \in Ab\}_{n} of abelian groups together with group homomorphisms {∂n:Cn+1→Cn}\{\partial_n : C_{n+1} \to C_n\} such that ∂∘∂=0\partial \circ \partial = 0.

Remark

Remark

More generally, for RR any unital ring one can form the degreewise free moduleR[SingX]R[Sing X] over RR. The corresponding homology is the singular homology with coefficients in RR, denoted Hn(X,R)H_n(X,R). This generality we come to below in the next section.

Definition

For C•(S)C_\bullet(S) a chain complex as in def. 24 and for n∈ℕn \in \mathbb{N}, the degree-nnchain homology groupHn(C(S))∈AbH_n(C(S)) \in Ab is the quotient group

In particular for each n∈ℕn \in \mathbb{N} singular homology extends to a functor

Hn(−):Top→Ab.
H_n(-) : Top \to Ab
\,.

We close this section by stating the basic properties of singular homology, which make precise the sense in which it is an abelian approximation to the homotopy type of XX. The proof of these statements requires some of the tools of homological algebra that we develop in the later chapters, as well as some tools in algebraic topology.

from the chain complex concentrated on the morphism of multiplication by 2 on integers, to the chain complex concentrated on the cyclic group of order 2.

This clearly induces an isomorphism on all homology groups. But there is not even a non-zero chain map in the other direction, since there is no non-zero group homomorphism ℤ/2ℤ→ℤ\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}.

But quasi-isomorphisms are a little coarser than weak homotopy equivalences. The singular chain functor C•(−)C_\bullet(-) forgets some of the information in the homotopy types of topological spaces. The following series of statements characterizes to some extent what exactly is lost when passing to singular homology, and which information is in fact retained.

Since a homotopy group in positive degree depends on the homotopy type of the connected component of the base point, while the singular homology does not depend on a basepoint, it is interesting to compare these groups only for the case that XX is connected.

3) Categories of chain complexes

In def. 24 we had encountered complexes of singular chains, of formal linear combinations of simplices in a topological space. Here we discuss such chain complexes in their own right in a bit more depth.

Definition

an object is a groupAA such that for all elements a1,a2∈Aa_1, a_2 \in A we have that the group product of a1a_1 with a2a_2 is the same as that of a2a_2 with a1a_1, which we write a1+a2∈Aa_1 + a_2 \in A (and the neutral element is denoted by 0∈A0 \in A);

a morphismϕ:A1→A2\phi : A_1 \to A_2 is a group homomorphism, hence a function of the underlying sets, such that for all elements as above ϕ(a1+a2)=ϕ(a1)+ϕ(a2)\phi(a_1 + a_2) = \phi(a_1) + \phi(a_2).

Notice that this is not a group homomorphism out of the product group. The product group A×BA \times B is the group whose elements are pairs (a,b)(a,b) with a∈Aa \in A and b∈Bb \in B, and whose group operation is

there is an associativitynatural isomorphism(A⊗B)⊗C→≃A⊗(B⊗C)(A \otimes B) \otimes C \stackrel{\simeq}{\to} A \otimes (B \otimes C) which is “coherent” in the sense that all possible ways of using it to rebracket a given expression are equal.

There is a unitnatural isomorphismA⊗ℤ→≃AA \otimes \mathbb{Z} \stackrel{\simeq}{\to} A which is compatible with the asscociativity isomorphism in the evident sense.

Proof

To see that ℤ\mathbb{Z} is the unit object, consider for any abelian group AA the map

This shows that A⊗ℤ→AA \otimes \mathbb{Z} \to A is in fact an isomorphism.

The other properties are similarly direct to check.

We see simple but useful examples of tensor products of abelian groups put to work below in the context of example 29 and then in many of the applications to follow. An elementary but not entirely trivial example that may help to illustrate the nature of the tensor product is the following.

Just as an outlook and a suggestion for how to think geometrically of the objects appearing here, we mention the following.

Remark

The Gelfand duality theorem says that if one remembers certain extra structure on the rings of functions C(X,ℂ)C(X, \mathbb{C}) in example 16 – called the structure of a C-star algebra, then this construction

is an equivalence of categories between that of topological spaces, and the opposite category of C*C^\ast-algebras. Together with remark 13 further below this provides a useful dual geometric way of thinking about the theory of modules.

Example

Given a complex vector bundleE→XE \to X on XX, write Γ(E)\Gamma(E) for its set of continuous sections. Since for each point x∈Xx \in X the fiberExE_x of EE over xx is a ℂ\mathbb{C}-module (by example 21), Γ(X)\Gamma(X) is a C(X,ℂ)C(X,\mathbb{C})-module.

Just as an outlook and a suggestion for how to think of modules geometrically, we mention the following.

We now discuss a bunch of properties of the category RRMod which together will show that there is a reasonable concept of chain complexes of RR-modules, in generalization of how there is a good concept of chain complexes of abelian groups. In a more abstract category theoretical context than we invoke here, all of the following properties are summarized in the following statement.

Definition

Remark

This means that 0∈𝒞0 \in \mathcal{C} is a zero object precisely if for every other object AA there is a unique morphismA→0A \to 0 to the zero object as well as a unique morphism 0→A0 \to A from the zero object.

Proposition

Proof

Clearly the 0-module 00 is a terminal object, since every morphism N→0N \to 0 has to send all elements of NN to the unique element of 00, and every such morphism is a homomorphism. Also, 0 is an initial object because a morphism 0→N0 \to N always exists and is unique, as it has to send the unique element of 0, which is the neutral element, to the neutral element of NN.

Remark

for every object CC and every morphism h:C→Ah : C \to A such that f∘h=0f\circ h = 0 is the zero morphism, there is a unique morphism ϕ:C→ker(f)\phi : C \to ker(f) such that h=p∘ϕh = p\circ \phi.

Example

In the categoryAb of abelian groups, the kernel of a group homomorphismf:A→Bf : A \to B is the subgroup of AA on the set f−1(0)f^{-1}(0) of elements of AA that are sent to the zero-element of BB.

Example

More generally, for RR any ring, this is true in RRMod: the kernel of a morphism of modules is the preimage of the zero-element at the level of the underlying sets, equipped with the unique sub-module structure on that set.

Proof

Suppose that ff is a monomorphism, hence that f:N1→N2f : N_1 \to N_2 is such that for all morphisms g1,g2:K→N1g_1, g_2 : K \to N_1 such that f∘g1=f∘g2f \circ g_1 = f \circ g_2 already g1=g2g_1 = g_2. Let then g1g_1 and g2g_2 be the inclusion of submodules generated by a single element k1∈Kk_1 \in K and k2∈Kk_2 \in K, respectively. It follows that if f(k1)=f(k2)f(k_1) = f(k_2) then already k1=k2k_1 = k_2 and so ff is an injection. Conversely, if ff is an injection then its image is a submodule and it follows directly that ff is a monomorphism.

Suppose now that ff is an epimorphism and hence that f:N1→N2f : N_1 \to N_2 is such that for all morphisms g1,g2:N2→Kg_1, g_2 : N_2 \to K such that f∘g1=f∘g2f \circ g_1 = f \circ g_2 already g1=g2g_1 = g_2. Let then g1:N2→N2im(f)g_1 : N_2 \to \frac{N_2}{im(f)} be the natural projection. and let g2:N2→0g_2 : N_2 \to 0 be the zero morphism. Since by construction f∘g1=0f \circ g_1 = 0 and f∘g2=0f \circ g_2 = 0 we have that g1=0g_1 = 0, which means that Nim(f)=0\frac{N}{im(f)} = 0 and hence that N=im(f)N = im(f) and so that ff is surjective. The other direction is evident on elements.

Definition

For N1,N2∈RModN_1, N_2 \in R Mod two modules, define on the hom setHomRMod(N1,N2)Hom_{R Mod}(N_1,N_2) the structure of an abelian group whose addition is given by argumentwise addition in N2N_2: (f1+f2):n↦f1(n)+f2(n)(f_1 + f_2) : n \mapsto f_1(n) + f_2(n).

Proof

Linearity of composition in the second argument is immediate from the pointwise definition of the abelian group structure on morphisms. Linearity of the composition in the first argument comes down to linearity of the second module homomorphism.

Remark

In fact RModR Mod is even a closed category, but this we do not need for showing that it is abelian.

Corollary

Proposition

The products are given by cartesian product of the underlying sets with componentwise addition and RR-action.

The direct sum is the subobject of the product consisting of tuples of elements such that only finitely many are non-zero.

Proof

The defining universal properties are directly checked. Notice that the direct product ∏i∈INi\prod_{i \in I} N_i consists of arbitrary tuples because it needs to have a projection map

pj:∏i∈INi→Nj
p_j : \prod_{i \in I} N_i \to N_j

to each of the modules in the product, reproducing all of a possibly infinite number of non-trivial maps {K→Nj}\{K \to N_j\}. On the other hand, the direct sum just needs to contain all the modules in the sum

ιj:Nj→⊕i∈INi
\iota_j : N_j \to \oplus_{i \in I} N_i

and since, being a module, it needs to be closed only under addition of finitely many elements, so it consists only of linear combinations of the elements in the NjN_j, hence of finite formal sums of these.

Corollary

Proposition

Proof

Using prop. 18 this is directly checked on the underlying sets: given a monomorphism K↪NK \hookrightarrow N, its cokernel is N→NKN \to \frac{N}{K}, The kernel of that morphism is evidently K↪NK \hookrightarrow N.

Now we finally have all the ingredients to talk about chain complexes of RR-modules. The following definitions are the direct analogs of the definitions of chain complexes of abelian groups in Simplicial and singular homology above.

Corollary

This establishes the basic objects that we are concerned with in the following. But as before, we are not so much interested in chain complexes up to chain map isomorphism, rather, we are interested in them up to a notion of homotopy equivalence. This we begin to study in the next section Homology exact sequences and homotopy fiber sequences. But in order to formulate that neatly, it is useful to have the tensor product of chain complexes. We close this section with introducing that notion.

Definition

For X,Y∈Ch•(𝒜)X, Y \in Ch_\bullet(\mathcal{A}) write X⊗Y∈Ch•(𝒜)X \otimes Y \in Ch_\bullet(\mathcal{A}) for the chain complex whose component in degree nn is given by the direct sum

where in the last line we express a general element as a linear combination of the canonical basis elements which are obtained as tensor products (a,b)∈R⊗R(a,b) \in R\otimes R of the previous basis elements. Notice that by the definition of tensor product of modules we have relations like

says that the oriented boundary of the bottom morphism is the bottom right element (its target) minus the bottom left element (its source), as indicated. Here we used that the differential of a degree-0 element in I•I_\bullet is 0, and hence so is any tensor product with it.

which can be read as saying that the boundary is the evident boundary thought of as oriented by drawing it counterclockwise into the plane, so that the right arrow (which points up) contributes with a +1 prefactor, while the left arrow (which also points up) contributes with a -1 prefactor.

Definition

One usually writes this just “0→A→B→C→00 \to A \to B \to C \to 0” or even just “A→B→CA \to B \to C”.

Remark

A general exact sequence is sometimes called a long exact sequence, to distinguish from the special case of a short exact sequence.

Beware that there is a difference between A→B→CA \to B \to C being exact (at BB) and A→B→CA \to B \to C being a “short exact sequence” in that 0→A→B→C→00 \to A \to B \to C \to 0 is exact at AA, BB and CC. This is illustrated by the following proposition.

Proof

The third condition is the definition of exactness at BB. So we need to show that the first two conditions are equivalent to exactness at AA and at CC.

This is easy to see by looking at elements when 𝒜≃R\mathcal{A} \simeq RMod, for some ring RR (and the general case can be reduced to this one using one of the embedding theorems):

The sequence being exact at

0→A→B
0 \to A \to B

means, since the image of 0→A0 \to A is just the element 0∈A0 \in A, that the kernel of A→BA \to B consists of just this element. But since A→BA \to B is a group homomorphism, this means equivalently that A→BA \to B is an injection.

Dually, the sequence being exact at

B→C→0
B \to C \to 0

means, since the kernel of C→0C \to 0 is all of CC, that also the image of B→CB \to C is all of CC, hence equivalently that B→CB \to C is a surjection.

Example

Let 𝒜=ℤ\mathcal{A} = \mathbb{Z}Mod≃\simeqAb. For n∈ℕn \in \mathbb{N} with n≥1n \geq 1 let ℤ→⋅nℤ\mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z} be the linear map/homomorphism of abelian groups which acts by the ordinary multiplication of integers by nn. This is clearly an injection. The cokernel of this morphism is the projection to the quotient group, which is the cyclic groupℤn≔ℤ/nℤ\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}. Hence we have a short exact sequence

Proposition

Proof

It is clear that the third condition implies the first two: take the section/retract to be given by the canonical injection/projection maps that come with a direct sum.

Conversely, suppose we have a retract r:B→Ar \colon B \to A of i:A→Bi \colon A \to B. Write P:B→rA→iBP \colon B \stackrel{r}{\to} A \stackrel{i}{\to} B for the composite. Notice that by r∘i=idr\circ i = id this is an idempotent: P∘P=PP \circ P = P, hence a projector.

Then every element b∈Bb \in B can be decomposed as b=(b−P(b))+P(b)b = (b - P(b)) + P(b) hence with b−P(b)∈ker(r)b - P(b) \in ker(r) and P(b)∈im(i)P(b) \in im(i). Moreover this decomposition is unique since if b=i(a)b = i(a) while at the same time r(b)=0r(b) = 0 then 0=r(i(a))=a0 = r(i(a)) = a. This shows that B≃im(i)⊕ker(r)B \simeq im(i) \oplus ker(r) is a direct sum and that i:A→Bi \colon A \to B is the canonical inclusion of im(i)im(i). By exactness it then follows that ker(r)≃ker(p)ker(r) \simeq ker(p) and hence that B≃A⊕CB \simeq A \oplus C with the canonical inclusion and projection.

The implication that the second condition also implies the third is formally dual to this argument.

Moreover, of particular interest are exact sequences of chain complexes. We consider this concept in full beauty below in section 5). In order to motivate the discussion there we here content ourselves with the following quick definition, which already admits discussion of some of its rich consequences.

To see that δn\delta_n is a group homomorphism, let [c]=[c1]+[c2][c] = [c_1] + [c_2] be a sum. Then c^≔c^1+c^2\hat c \coloneqq \hat c_1 + \hat c_2 is a lift and by linearity of ∂\partial we have [∂Bc^]A=[∂Bc^1]+[∂Bc^2][\partial^B \hat c]_A = [\partial^B \hat c_1] + [\partial^B \hat c_2].

Proof

Consider first the exactness of Hn(A)→Hn(i)Hn(B)→Hn(p)Hn(C)H_n(A) \stackrel{H_n(i)}{\to} H_n(B)
\stackrel{H_n(p)}{\to} H_n(C).

It is clear that if a∈Zn(A)↪Zn(B)a \in Z_n(A) \hookrightarrow Z_n(B) then the image of [a]∈Hn(B)[a] \in H_n(B) is [p(a)]=0∈Hn(C)[p(a)] = 0 \in H_n(C). Conversely, an element [b]∈Hn(B)[b] \in H_n(B) is in the kernel of Hn(p)H_n(p) if there is c∈Cn+1c \in C_{n+1} with ∂Cc=p(b)\partial^C c = p(b). Since pp is surjective let c^∈Bn+1\hat c \in B_{n+1} be any lift, then [b]=[b−∂Bc^][b] = [b - \partial^B \hat c] but p(b−∂Bc)=0p(b - \partial^B c) = 0 hence by exactness b−∂Bc^∈Zn(A)↪Zn(B)b - \partial^B \hat c \in Z_n(A) \hookrightarrow Z_n(B) and so [b][b] is in the image of Hn(A)→Hn(B)H_n(A) \to H_n(B).

It remains to see that

the image of Hn(B)→Hn(C)H_n(B) \to H_n(C) is the kernel of δn\delta_n;

the kernel of Hn−1(A)→Hn−1(B)H_{n-1}(A) \to H_{n-1}(B) is the image of δn\delta_n.

This follows by inspection of the formula in def. 54. We spell out the first one:

If [c][c] is in the image of Hn(B)→Hn(C)H_n(B) \to H_n(C) we have a lift c^\hat c with ∂Bc^=0\partial^B \hat c = 0 and so δn[c]=[∂Bc^]A=0\delta_n[c] = [\partial^B \hat c]_A = 0. Conversely, if for a given lift c^\hat c we have that [∂Bc^]A=0[\partial^B \hat c]_A = 0 this means there is a∈Ana \in A_n such that ∂Aa≔∂Ba=∂Bc^\partial^A a \coloneqq \partial^B a = \partial^B \hat c. But then c˜≔c^−a\tilde c \coloneqq \hat c - a is another possible lift of cc for which ∂Bc˜=0\partial^B \tilde c = 0 and so [c][c] is in the image of Hn(B)→Hn(C)H_n(B) \to H_n(C).

Example

We now discuss a deeper, more conceptual way of understanding the origin of long exact sequences in homology and the nature of connecting homomorphisms. This will give first occasion to see some actual homotopy theory of chain complexes at work, and hence serves also as a motivating example for the discussions to follow in chapter III).

For this we need the notion of chain homotopy, which is the abelianized analog of the notion of homotopy of continuous maps above in def. 8. We now first introduce this concept by straightforwardly mimicking the construction in def. 8 with topological spaces replaced by chain complexes. Then we use chain homotopies to construct mapping cones of chain maps. Finally we explain how these refine the above long exact sequences in homology groups to homotopy cofiber sequences of the chain complexes themselves.

A chain homotopy is a homotopy in Ch•(𝒜)Ch_\bullet(\mathcal{A}). We first give the explicit definition, the more abstract characterization is below in prop. 33.

Definition

A chain homotopyψ:f⇒g\psi : f \Rightarrow g between two chain mapsf,g:C•→D•f,g : C_\bullet \to D_\bullet in Ch•(𝒜)Ch_\bullet(\mathcal{A}) is a sequence of morphisms

where the term ℤ⊕ℤ\mathbb{Z} \oplus \mathbb{Z} is in degree 0: this is the free abelian group on the set {(0),(1)}\{(0),(1)\} of 0-simplices in Δ[1]\Delta[1]. The other copy of ℤ\mathbb{Z} is the free abelian group on the single non-degenerate edge (0→1)(0 \to 1) in Δ[1]\Delta[1]. (All other simplices of Δ[1]\Delta[1] are degenerate and hence do not contribute to the normalized chain complex which we are discussing here.) The single nontrivial differential sends 1∈ℤ1 \in \mathbb{Z} to (−1,1)∈ℤ⊕ℤ(-1,1) \in \mathbb{Z} \oplus \mathbb{Z}, reflecting the fact that one of the vertices is the 0-boundary the other the 1-boundary of the single nontrivial edge.

Therefore a chain map (f,g,ψ):I•⊗C•→D•(f,g,\psi) : I_\bullet \otimes C_\bullet \to D_\bullet that restricted to the two copies of C•C_\bullet is ff and gg, respectively, is characterized by a collection of commuting diagrams

On the elements (1,0,0)(1,0,0) and (0,1,0)(0,1,0) in the top left this reduces to the chain map condition for ff and gg, respectively. On the element (0,0,1)(0,0,1) this is the equation for the chain homotopy

Remark

Beware, as we will discuss in detail below in 8), that another category that would deserve to carry this name instead is called the derived category of 𝒜\mathcal{A}. In the derived category one also quotients out chain homotopy, but one allows that first the domain of the two chain maps ff and gg is refined along a quasi-isomorphism.

Remark

Quasi-isomorphisms are also called, more descriptively, homology isomorphisms or H•H_\bullet-isomorphisms. See at homology localization for more on this.

With the homotopy theoretic notions of chain homotopy and quasi-isomorphism in hand, we can now give a deeper explanation of long exact sequences in homology. We first give now a heuristic discussion that means to serve as a guide through the constructions to follow. The reader wishing to skip this may directly jump ahead to definition 38.

For if a chain mapA•→B•A_\bullet \to B_\bullet is the degreewise kernel of a chain map B•→C•B_\bullet \to C_\bullet, then if A^•→≃A•\hat A_\bullet \stackrel{\simeq}{\to} A_\bullet is a quasi-isomorphism (for instance a projective resolution of A•A_\bullet) then of course the composite chain map A^•→B•\hat A_\bullet \to B_\bullet is in general far from being the degreewise kernel of C•C_\bullet. Hence the notion of degreewise kernels of chain maps and hence that of short exact sequences is not meaningful in the homotopy theory of chain complexes in 𝒜\mathcal{A} (for instance: not in the derived category of 𝒜\mathcal{A}).

That short exact sequences of chain complexes nevertheless play an important role in homological algebra is due to what might be called a “technical coincidence”:

is not only a pushout square in Ch•(𝒜)Ch_\bullet(\mathcal{A}), exhibiting C•C_\bullet as the cofiber of A•→B•A_\bullet \to B_\bullet over 0∈C•0 \in C_\bullet, it is in fact also a homotopy pushout.

But a central difference between fibers/cofibers on the one hand and homotopy fibers/homotopy cofibers on the other is that while the (co)fiber of a (co)fiber is necessarily trivial, the homotopy (co)fiber of a homotopy (co)fiber is in general far from trivial: it is instead the loopingΩ(−)\Omega(-) or suspensionΣ(−)\Sigma(-) of the codomain/domain of the original morphism: by the pasting law for homotopy pullbacks the pasting composite of successive homotopy cofibers of a given morphism f:A•→B•f : A_\bullet \to B_\bullet looks like this:

cone(f)cone(f) is a specific representative of the homotopy cofiber of ff called the mapping cone of ff, whose construction comes with an explicit chain homotopyϕ\phi as indicated, hence cone(f)cone(f) is homology-equivalence to C•C_\bullet above, but is in general a “bigger” model of the homotopy cofiber;

And applying the chain homology functor to this yields the long exact sequence in chain homology which is traditionally said to be associated to the short exact sequence A•→B•→C•A_\bullet \to B_\bullet \to C_\bullet.

In conclusion this means that it is not really the passage to homology groups which “makes a short exact sequence become long”. It’s rather that passing to homology groups is a shadow of passing to chain complexes regarded up to quasi-isomorphism, and this is what makes every short exact sequence be realized as but a special presentation of a stage in a long homotopy fiber sequence.

In the following we repeatedly mention that certain chain complexes are colimits of certain diagrams of chain complexes. The reader unfamiliar with colimits may simply ignore them and regard the given chain complex as arising by definition. However, even a vague intuitive understanding of the indicated colimits as formalizations of “gluing” of chain complexes along certain maps should help to motivate why these definitions are what they are. The reader unhappy even with this can jump ahead to prop. 40 and take this and the following propositions up to and including prop. 43 as definitions.

The notion of a mapping cone that we introduce now is something that makes sense whenever

there is a notion of cylinder object, such as the topological cylinder [0,1]×X[0,1] \times X over a topological space, or the chain complex cylinder I•⊗X•I_\bullet \otimes X_\bullet of a chain complex from def. 56.

there is a way to glue objects along maps between them, a notion of colimit.

Heuristically it is clear that this way every cycle in YY that happens to be in the image of XX can be “continuously” translated in the cylinder-direction, keeping it constant in YY, to the other end of the cylinder, where it becomes the point. This means that every homotopy group of YY in the image of ff vanishes in the mapping cone. Hence in the mapping cone the image of XX under ff in YY is removed up to homotopy. This makes it clear how cone(f)cone(f) is a homotopy-version of the cokernel of ff. And therefore the name “mapping cone”.

Another interpretation of the mapping cone is just as important:

Remark

A morphism η:cyl(X)→Y\eta : cyl(X) \to Y out of a cylinder object is a left homotopyη:g⇒h\eta : g \Rightarrow h between its restrictions g≔η(0)g\coloneqq \eta(0) and h≔η(1)h \coloneqq \eta(1) to the cylinder boundaries

defines the mapping conecone(f)cone(f) of ff: the result of forming the cyclinder over XX and then identifying one end with the point and the other with YY, via ff.

Remark

As in remark 23 all these step have evident heuristic geometric interpretations:

cone(X)cone(X) is obtained from the cylinder over XX by contracting one end of the cylinder to the point;

cyl(f)cyl(f) is obtained from the cylinder over XX by gluing YY to one end of the cylinder, as specified by the map ff;

We discuss now this general construction of the mapping cone cone(f)cone(f) for a chain mapff between chain complexes. The end result is prop. 43 below, reproducing the classical formula for the mapping cone.

Proof

This follows by starting with remark 28 and then following these inclusions through the formation of the two colimits as discussed above.

Using these mapping cones of chain maps, we now explain how the long exact sequences of homology groups, prop. 32, are a shadow under homology of genuine homotopy cofiber sequences of the chain complexes themselves.

Let f:X•→Y•f : X_\bullet \to Y_\bullet be a chain map and write cone(f)∈Ch•(𝒜)cone(f) \in Ch_\bullet(\mathcal{A}) for its mapping cone as explicitly given in prop. 43.

is given, via prop. 43, by the canonical projection out of a direct sum

pn:Yn⊕Xn−1→Xn−1.
p_n : Y_\n \oplus X_{n-1} \to X_{n-1}
\,.

This defines the mapping cone construction on chain complex. Its definition as a universal left homotopy should make the following proposition at least plausible, which we cannot prove yet at this point, but which we state nevertheless to highlight the meaning of the mapping cone construction. The tools for the proof of propositions like this are discussed further below in 7) Derived categories and derived functors.

since the composite morphism is the inclusion of YY followed by the bottom morphism on YY.

Abstractly, this already implies that cone(f)•→Z•cone(f)_\bullet \to Z_\bullet is a quasi-isomorphism, for this diagram gives a morphism of cocones under the diagram defining cone(f)cone(f) in prop. 38 and by the above both of these cocones are homotopy-colimiting.

But in checking the claimed inverse of the induced map on homology groups, we verify this also explicity:

We first determine those cycles (xn−1,yn)∈cone(f)n(x_{n-1}, y_n) \in cone(f)_n which lift a cycle znz_n. By lemma 38 a lift of chains is any pair of the form (xn−1,z^n)(x_{n-1}, \hat z_n) where z^n\hat z_n is a lift of znz_n through Yn→XnY_n \to X_n. So xn−1x_{n-1} has to be found such that this pair is a cycle. By prop. 43 the differential acts on it by

xn−1≔−∂Yz^nx_{n-1} \coloneqq -\partial^Y \hat z_n (which implies ∂Xxn−1=−∂X∂Yz^n=−∂Y∂Yz^n=0\partial^X x_{n-1} = -\partial^X \partial^Y \hat z_n = -\partial^Y \partial^Y \hat z_n = 0 due to the fact that fnf_n is assumed to be an inclusion, hence that ∂X\partial^X is the restriction of ∂Y\partial^Y to elements in XnX_n).

This condition clearly has a unique solution for every lift z^n\hat z_n and a lift z^n\hat z_n always exists since pn:Yn→Znp_n : Y_n \to Z_n is surjective, by assumption that we have a short exact sequence of chain complexes. This shows that Hn(h•)H_n(h_\bullet) is surjective.

To see that it is also injective we need to show that if a cycle(−∂Yz^n,z^n)∈cone(f)n(-\partial^Y \hat z_n, \hat z_n) \in cone(f)_n maps to a cycle zn=pn(z^n)z_n = p_n(\hat z_n) that is trivial in Hn(Z)H_n(Z) in that there is cn+1c_{n+1} with ∂Zcn+1=zn\partial^Z c_{n+1} = z_n, then also the original cycle was trivial in homology, in that there is (xn,yn+1)(x_n, y_{n+1}) with

By prop. 1 the inverse of the vertical map is given by choosing lifts and forming the corresponding element given by the connecting homomorphism. By prop. 45 the horizontal map is just the projection, and hence the assignment is of the form

6) Double complexes and the diagram chasing lemmas

We have seen in the discussion of the connecting homomorphism in the homology long exact sequence in 4) above that given an exact sequence of chain complexes – hence in particular a chain complex of chain complexes – there are interesting ways to relate elements on the far right to elements on the far left in lower degree. In 5) we had given the conceptual explanation of this phenomenon in terms of long homotopy fiber sequences. But often it is just computationally useful to be able to efficiently establish and compute these “long diagram chase”-relations, independently of a homotopy-theoretic interpretation. Such computational tools we discuss here.

A chain complex of chain complex is called a double complex and so we first introduce this elementary notion and the corresponding notion notion of total complex. (Total complexes are similarly elementary to define but will turn out to play a deeper role as models for homotopy colimits, this we indicate further below in chapter V)).

There is a host of classical diagram-chasing lemmas that relate far-away entries in double complexes that enjoy suitable exactness properties. These go by names such as the snake lemma or the 3x3 lemma. The underlying mechanism of all these lemmas is made most transparent in the salamander lemma. This is fairly trivial to establish, and the notions it induces allow quick transparent proofs of all the other diagram-chasing lemmas.

III) Abelian homotopy theory

We have seen in section II) that the most interesting properties of the category of chain complexes is all secretly controled by the phenomenon of chain homotopy and quasi-isomorphism. Strictly speaking these two phenomena point beyond plain category theory to the richer context of general abstract homotopy theory. Here we discuss properties of the category of chain complexes from this genuine homotopy-theoretic point of view. The result of passing the category of chain complexes to genuine homotopy theory is called the derived category (of the underlying abelian category𝒜\mathcal{A}, say of modules) and we start in 7) with a motivation of the phenomenon of this “homotopy derivation” and the discussion of the necessary resolutions of chain complexes. This naturally gives rise to the general notion of derived functors which we discuss in 8). Examples of these are ubiquituous in homological algebra, but as in ordinary enriched category theory two stand out as being of more fundamental importance, the derived functor “Ext” of the hom-functor and the derived functor “Tor” of the tensor product functor. Their properties and uses we discuss in 9).

7) Chain homotopy and resolutions

We now come back to the category𝒦(𝒜)\mathcal{K}(\mathcal{A}) of def. 59, the “homotopy category of chain complexes” in which chain-homotopic chain maps are identified. This would seem to be the right context to study the homotopy theory of chain complexes, but one finds that there are still chain maps which ought to be identified in homotopy theory, but which are still not identified in 𝒦(𝒜)\mathcal{K}(\mathcal{A}). This is our motivating example 33 below.

We discuss then how this problem is fixed by allowing to first “resolve” chain complexes quasi-isomorphically by “good representatives” called projective resolutions or injective resolutions. Many of the computations in the following sections – and in homological algebra in general – come down to operating on such resolutions. We end this section by prop. 53 below, which shows that the above problem indeed goes away when allowing chain complexes to be resolved.

In the next section, 8), we discuss how this process of forming resolutions functorially extends to the whole category of modules.

So we start here with this simple example that shows the problem with bare chain homotopies and indicates how these have to be resolved:

Example

In Ch•(𝒜)Ch_\bullet(\mathcal{A}) for 𝒜=\mathcal{A} = Ab consider the chain map

But the above chain map is chain homotopic precisely only to itself. This is because the degree-0 component of any chain homotopy out of this has to be a homomorphism of abelian groups ℤ2→ℤ\mathbb{Z}_2 \to \mathbb{Z}, and this must be the 0-morphism, because ℤ\mathbb{Z} is a free group, but ℤ2\mathbb{Z}_2 is not.

This points to the problem: the components of the domain chain complex are not free enough to admit sufficiently many maps out of it.

So resolving the domain by a sufficiently free complex makes otherwise missing chain homotopies exist. Below in lemma 5 we discuss the general theory behind the kind of situation of this example. But to get there we first need some basic notions and facts.

Notably, in general it is awkward to insist on actual free resolutions. But it is easy to see, and this we discuss now, that essentially just as well is a resolution by modules which are direct summands of free modules.

Definition

Remark

The point of this lifting property will become clear when we discuss the construction of projective resolutions a bit further below: they are built by applying this property degreewise to obtain suitable chain maps.

We will be interested in projective objects in the category RRMod: projective modules. Before we come to that, notice the following example (which the reader may on first sight feel is pedantic and irrelevant, but for the following it is actually good to make this explicit).

This is a major aspect of homological algebra. While we will not discuss this further here in this introduction, the reader might enjoy keeping in mind that all of the following discussion of resolutions of RR-modules goes through in this wider context of sheaves of modules except for subtleties related to the (partial) failure of example 34 for the category of sheaves.

We now characterize projective modules.

Lemma

Proof

Explicitly: if S∈SetS \in Set and F(S)=R(S)F(S) = R^{(S)} is the free module on SS, then a module homomorphism F(S)→NF(S) \to N is specified equivalently by a functionf:S→U(N)f : S \to U(N) from SS to the underlying set of NN, which can be thought of as specifying the images of the unit elements in R(S)≃⊕s∈SRR^{(S)} \simeq \oplus_{s \in S} R of the |S|{\vert S\vert} copies of RR.

Accordingly then for N˜→N\tilde N \to N an epimorphism, the underlying function U(N˜)→U(N)U(\tilde N) \to U(N) is an epimorphism, and the axiom of choice in Set says that we have all lifts f˜\tilde f in

This is clearly an epimorphism. Thefore if NN is projective, there is a sectionss of ϵ\epsilon. This exhibits NN as a direct summand of F(U(N))F(U(N)).

We discuss next how to build resolutions of chain complexes by projective modules. But before we come to that it is useful to also introduce the dual notion. So far we have concentrated on chain complexes with degrees in the natural numbers: non-negative degrees. For a discussion of resolutions we need a more degree-symmetric perspective, which of course is straightforward to obtain.

Definition

A cochain complexC•C^\bullet in 𝒜=RMod\mathcal{A} = R Mod is a sequence of morphism

Example

Let N∈𝒜N \in \mathcal{A} be a fixed module and C•∈Ch•(𝒜)C_\bullet \in Ch_\bullet(\mathcal{A}) a chain complex. Then applying degreewise the hom-functor out of the components of C•C_\bullet into NN yields a cochain complex in ℤMod≃\mathbb{Z} Mod \simeq Ab:

Example

The group of rational numbersℚ\mathbb{Q} is injective in Ab, as is the additive group of real numbersℝ\mathbb{R} and generally that underlying any field. The additive group underlying any vector space is injective. The quotient of any injective group by any other group is injective.

Proof

To start with, notice that the group ℚ\mathbb{Q} of rational numbers is divisible and hence the canonical embedding ℤ↪ℚ\mathbb{Z} \hookrightarrow \mathbb{Q} shows that the additive group of integers embeds into an injective ℤ\mathbb{Z}-module.

Here A˜\tilde A is divisible because the direct sum of divisible groups is again divisible, and also the quotient group of a divisible groups is again divisble. So this exhibits an embedding of any AA into a divisible abelian group, hence into an injective ℤ\mathbb{Z}-module.

By the assumption that there are enough injectives in 𝒜\mathcal{A} we may now again find a monomorphism Jn/Jn−1↪iJn+1J^n/J^{n-1} \stackrel{i}{\hookrightarrow} J^{n+1} into an injective object Jn+1J^{n+1}. This being a monomorphism means that

By the assumption that there are enough projectives in 𝒜\mathcal{A} we may now again find an epimorphism p:Jn+1→ker(∂n−1) p : J_{n+1} \to ker(\partial_{n-1}) out of a projective object Jn+1J_{n+1}. This being an epimorphism means that

To conclude this section we now show that all this work indeed serves to solve the problem indicated above in example 33.

Proposition

Let f•:X•→J•f^\bullet : X^\bullet \to J^\bullet be a chain map of cochain complexes in non-negative degree, out of an exact complex0≃qiX•0 \simeq_{qi} X^\bullet to a degreewise injective complex J•J^\bullet. Then there is a null homotopy

η:0⇒f•
\eta : 0 \Rightarrow f^\bullet

Proof

By definition of chain homotopy we need to construct a sequence of morphisms (ηn+1:Xn+1→Jn)n∈ℕ(\eta^{n+1} : X^{n+1} \to J^{n})_{n \in \mathbb{N}} such that

Proof

Hence we have seen now that injective and projective resolutions of chain complexes serve to make chain homotopy interact well with quasi-isomorphism. In the next section we show that this construction lifts from single chain complexes to chain maps between chain complexes and in fact to the whole category of chain complexes. The resulting “resolved” category of chain complexes is the derived category, the true home of the abelian homotopy theory of chain complexes.

8) The derived category

In the previous section we have seen that every object A∈𝒜A \in \mathcal{A} admits an injective resolution and a projective resolution. Here we lift this construction to morphisms and then to the whole category of chain complexes, up to chain homotopy.

The following proposition says that, when injectively resolving objects, the morphisms between these objects lift to the resolutions, and the following one, prop. 55, says that this lift is unique up to chain homotopy.

Assume then that for some n∈ℕn \in \mathbb{N} component maps f•≤nf^{\bullet \leq n} have been obtained such that dYk∘fk=fk+1∘dXkd^k_Y\circ f^k = f^{k+1}\circ d^k_X for all 0≤k<n0 \leq k \lt n . In order to construct fn+1f^{n+1} consider the following diagram, which we will describe/construct stepwise from left to right:

and therefore gng^n factors through Xn/im(dXn−1)X^n/im(d^{n-1}_X) via some hnh^n as indicated in the middle of the above diagram. Finally the morphism on the top right is a monomorphism by the fact that X•X^{\bullet} is exact in positive degrees (being quasi-isomorphic to a complex concentrated in degree 0) and so a lift fn+1f^{n+1} as shown on the far right of the diagram exists by the defining lifting property of the injective object Yn+1Y^{n+1}.

The total outer diagram now commutes, being built from commuting sub-diagrams, and this is the required chain map property of f•≤n+1f^{\bullet \leq n+1} This completes the induction step.

Proposition

The morphism f•f_\bullet in prop. 54 is the unique one up to chain homotopy making the given diagram commute.

Proof

Given two cochain maps g1•,g2•g_1^\bullet, g_2^\bullet making the diagram commute, a chain homotopyg1•⇒g2•g_1^\bullet \Rightarrow g_2^\bullet is equivalently a null homotopy0⇒g2•−g1•0 \Rightarrow g_2^\bullet - g_1^\bullet of the difference, which sits in a square of the form

where the second square from the left commutes due to the commutativity of the original square of chain complexes in degree 0.

Since h•h^\bullet is a quasi-isomorphism, the top chain complex is exact, by remark 32. Moreover the bottom complex consists of injective objects from the second degree on (the former degree 0). Hence the induction in the proof of prop. 53 implies the existence of a null homotopy

starting with η−1=0\eta^{-1} = 0 and η0=0\eta^{0 } = 0 (notice that the proof prop. 53 was formulated exactly this way), which works because f−1=0f^{-1} = 0. The de-augmentation {f•≥0}\{f^{\bullet \geq 0}\} of this is the desired null homotopy of f•f^\bullet.

We now discuss how the injective/projective resolutions constructed above are functorial if regarded in the homotopy category of chain complexes, def. 59. For definiteness, to be able to distinguish chain complexes from cochain complexes, introduce the following notation.

Definition

(the derived category)

Write as before

𝒦•(𝒜)∈Cat
\mathcal{K}_\bullet(\mathcal{A}) \in Cat

for the strong chain homotopy category of chain complexes, from def. 59.

These subcategories – or any category equivalent to them – are called the (strictly bounded above/below) derived category of 𝒜\mathcal{A}.

Remark

Often one defines the derived category by more general abstract means than we have introduced here, namely as the localization of the category of chain complexes at the quasi-isomorphims. If one does this, then the simple definition def. 74 is instead a theorem. The interested reader can find more details and further pointers here.

Proof

By prop. 51 every object X•∈Ch•(𝒜)X^\bullet \in Ch^\bullet(\mathcal{A}) has an injective resolution. Proposition 54 says that for X→X•X \to X^\bullet and X→X˜•X \to \tilde X^\bullet two resolutions there is a morphism X•→X˜•X^\bullet \to \tilde X^\bullet in 𝒦•(𝒜)\mathcal{K}^\bullet(\mathcal{A}) and prop. 55 says that this morphism is unique in 𝒦•(𝒜)\mathcal{K}^\bullet(\mathcal{A}). In particular it is therefore an isomorphism in 𝒦•(𝒜)\mathcal{K}^\bullet(\mathcal{A}) (since the composite with the reverse lifted morphism, also being unique, has to be the identity).

So choose one such injective resolution P(X)•P(X)^\bullet for each X•X^\bullet.

Then for f:X→Yf : X \to Y any morphism in 𝒜\mathcal{A}, proposition 51 again says that it can be lifted to a morphism between P(X)•P(X)^\bullet and P(Y)•P(Y)^\bullet and proposition 54 says that there is an image in 𝒦•(𝒜)\mathcal{K}^\bullet(\mathcal{A}), unique for morphism making the given diagram commute.

This implies that this assignment of morphisms is functorial, since then also the composites are unique.

For actually working with the derived category, the following statement is of central importance, which we record here without proof (which requires a bit of localization theory). It says that for computing hom-sets in the derived category, it is in fact sufficient to just resolve the domain or the codomain.

In conclusion we have found that there are resolution functors that embed 𝒜\mathcal{A} in the homotopically correct context of resolved chain complexes with chain maps up to chain homotopy between them.

In the next section we discuss the general properties of this “homotopically correct context”: the derived category.

etc. One wants to accordingly derive from FF a functor 𝒟•(𝒜)→𝒟•(𝒜)\mathcal{D}_\bullet(\mathcal{A}) \to \mathcal{D}_\bullet(\mathcal{A}) between these derived categories. It is immediate to achieve this on the domain category, there we can simply precompose and form

But the resulting composite lands in 𝒦•(𝒜′)\mathcal{K}_\bullet(\mathcal{A}') and in general does not factor through the inclusion 𝒟•(𝒜′)=𝒦•(𝒫𝒜′)↪𝒦•(𝒜′)\mathcal{D}_\bullet(\mathcal{A}') = \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}'}) \hookrightarrow \mathcal{K}_\bullet(\mathcal{A}').

In a more general abstract discussion than we present here, one finds that by applying a projective resolution functor on chain complexes, one can enforce this factorization. However, by definition of resolution, the resulting chain complex is quasi-isomorphic to the one obtained by the above composite.

This means that if one is only interested in the “weak chain homology type” of the chain complex in the image of a derived functor, then forming chain homology groups of the chain complexes in the images of the above composite gives the desired information. This is what def. 78 and def. 79 below do.

We record the following immediate consequence of this definition (which in the literature is often taken to be the definition).

Proposition

If FF is a left exact functor, then for every exact sequence of the form

0→A→B→C
0 \to A \to B \to C

also

0→F(A)→F(B)→F(C)
0 \to F(A) \to F(B) \to F(C)

is an exact sequence. Dually, if FF is a right exact functor, then for every exact sequence of the form

A→B→C→0
A \to B \to C \to 0

also

F(A)→F(B)→F(C)→0
F(A) \to F(B) \to F(C) \to 0

is an exact sequence.

Proof

If 0→A→B→C0 \to A \to B \to C is exact then A↪BA \hookrightarrow B is a monomorphism by prop. 28. But then the statement that A→B→CA \to B \to C is exact at BB says precisely that AA is the kernel of B→CB \to C. So if FF is left exact then by definition also F(A)→F(B)F(A) \to F(B) is the kernel of F(B)→F(C)F(B) \to F(C) and so is in particular also a monomorphism. Dually for right exact functors.

Remark

Proposition 57 is clearly the motivation for the terminology in def. 77: a functor is left exact if is preserves short exact sequences to the left, and right exact if it preserves them to the right.

Proof

By prop. 51 we can choose f•f_\bullet and h•h_\bullet. The task is now to construct the third resolution g•g_\bullet such as to obtain a short exact sequence of chain complexes, hence degreewise a short exact sequence, in the two row.

To construct this, let for each n∈ℕn \in \mathbb{N}

Bn≔An⊕Cn
B_n \coloneqq A_n \oplus C_n

be the direct sum and let the top horizontal morphisms be the canonical inclusion and projection maps of the direct sum.

which exists since C1C_1 is a projective object and A0→AA_0 \to A is an epimorphism by A•A_\bullet being a projective resolution. Here we are using that by exactness the bottom morphism indeed factors through AA as indicated, because the definition of ζ\zeta and the chain complex property of C•C_\bullet gives

This establishes g•g_\bullet such that the above diagram commutes and the bottom row is degreewise a short exact sequence, in fact a split exact sequence, by construction.

To see that g•g_\bullet is indeed a quasi-isomorphism, consider the homology long exact sequence associated to the short exact sequence of cochain complexes 0→A•→B•→C•→00 \to A_\bullet \to B_\bullet \to C_\bullet \to 0. In positive degrees it implies that the chain homology of B•B_\bullet indeed vanishes. In degree 0 it gives the short sequence 0→A→H0(B•)→B→00 \to A \to H_0(B_\bullet) \to B\to 0 sitting in a commuting diagram

Proof

of the given exact sequence which is itself again an exact sequence of cochain complexes.

Since AnA^n is an injective object for all nn, its component sequences 0→An→Bn→Cn→00 \to A^n \to B^n \to C^n \to 0 are indeed split exact sequences (see the discussion there). Splitness is preserved by any functor FF (and also since FF is additive it even preserves the direct sum structure that is chosen in the proof of lemma 6) and so it follows that

Proposition

Proof

Because an exact functor preserves all exact sequences. If Y•→AY_\bullet \to A is a projective resolution then also F(Y)•F(Y)_\bullet is exact in all positive degrees, and hence Ln≥1F(A))Hn≥(F(Y))=0L_{n\geq 1} F(A) ) H_{n \geq}(F(Y)) = 0. Dually for RnFR^n F.

Conversely:

Definition

Let F:𝒜→ℬF \colon \mathcal{A} \to \mathcal{B} be a left or right exact additive functor. An object A∈𝒜A \in \mathcal{A} is called an FF-acyclic object is all positive-degree right/left derived functors of FF are zero.

Acyclic objects are useful for computing derived functors on non-acyclic objects. More generally, we now discuss how the derived functor of an additive functor FF may also be computed not necessarily with genuine injective/projective resolutions, but with (just) “FF-injective”/“FF-projective resolutions”.

While projective resolutions in 𝒜\mathcal{A} are sufficient for computing everyleft derived functor on Ch•(𝒜)Ch_\bullet(\mathcal{A}) and injective resolutions are sufficient for computing everyright derived functor on Ch•(𝒜)Ch^\bullet(\mathcal{A}), if one is interested just in a single functor FF then such resolutions may be more than necessary. A weaker kind of resolution which is still sufficient is then often more convenient for applications. These FF-projective resolutions and FF-injective resolutions, respectively, we discuss now. A special case of both are FF-acyclic resolutions.

Example

if FF is left exact, then ℐ≔Ac\mathcal{I} \coloneqq Ac is a subcategory of FF-injective objects;

if FF is right exact, then 𝒫≔Ac\mathcal{P} \coloneqq Ac is a subcategory of FF-projective objects.

Proof

Consider the case that FF is right exact. The other case works dually. Then the first condition of def. 81 is satisfied because every injective object is an FF-acyclic object and by assumption there are enough of these.

and so on. Going by induction through this list and using the second condition in def. 81 we have that all the im(dn)im(d^n) are in ℐ\mathcal{I}. Then the third condition in def. 81 says that all the sequences

Theorem

For A∈𝒜A \in \mathcal{A} an object with FF-injective resolution A→≃qiIF•A \stackrel{\simeq_{qi}}{\to} I_F^\bullet, def. 83, we have for each n∈ℕn \in \mathbb{N} an isomorphism

RnF(A)≃Hn(F(IF•))
R^n F(A) \simeq H^n(F(I_F^\bullet))

between the nnth right derived functor, def. 78 of FF evaluated on AA and the cochain cohomology of FF applied to the FF-injective resolution IF•I_F^\bullet.

Proof

By prop. 51 we can also find an injective resolution A→≃qiI•A \stackrel{\simeq_{qi}}{\to} I^\bullet. By prop. 54 there is a lift of the identity on AA to a chain mapIF•→I•I^\bullet_F \to I^\bullet such that the diagram

Let Cone(f)∈Ch•(𝒜)Cone(f) \in Ch^\bullet(\mathcal{A}) be the mapping cone of ff and let I•→Cone(f)I^\bullet \to Cone(f) be the canonical chain map into it. By the explicit formulas for mapping cones, we have that

there is an isomorphismF(Cone(f))≃Cone(F(f))F(Cone(f)) \simeq Cone(F(f));

Observe that with f•f^\bullet a quasi-isomorphism Cone(f•)Cone(f^\bullet) is quasi-isomorphic to 0. Therefore the second item above implies with lemma 8 that also F(Cone(f))F(Cone(f)) is quasi-isomorphic to 0. This finally means that the above homology exact sequences consists of exact pieces of the form

This concludes the discussion of the general definition and the general properties of derived functors that we will consider here. In the next section we discuss the two archetypical examples.

10) Fundamental examples of derived functors

We introduce here the two archetypical examples of derived functors and discuss their basic properties. In the next chapter IV) The fundamental theorems we discuss how to use these derived functors for obtaining deeper statements.

For simplicity – this here being an introduction – we will discuss various statements only over R=ℤR = \mathbb{Z}, hence for abelian groups. The main simplification that this leads to is the following.

Proposition

This is a classical fact going back to Dedekind, now known (in its generalization to not-necessarily abelian groups) as the Nielsen-Schreier theorem. For us it is interesting due to the following consequence

Proof

By the proof of prop. 47 there is an epimorphismF0→AF_0 \to A out of a free abelian group (take for instance F0=F(U(A))F_0 = F(U(A)), the free abelian group in the underlying set of AA). By prop. 62 the kernel of this epimorphism is itself a free group, and hence by prop. 46 is itself projective. Take this kernel to be F1↪F0F_1 \hookrightarrow F_0.

This fact drastically constrains the complexity of right derived functors on abelian groups:

Proposition

Proof

By prop. 63 there is a projective resolution of any A∈AbA \in Ab of the form F•=[⋯→0→0→F1→F0]F_\bullet = [\cdots \to 0 \to 0 \to F_1 \to F_0]. This implies the claim by def. 78.

Remark

The conclusion of prop. 63 holds more generally over every ring which is a principal ideal domain. This includes in particular R=kR = k a field, in which case RMod≃kR Mod \simeq kVect. On the other hand, every kk-vector space is already projective itself, so that in this case the whole theory of right derived functors trivializes.

Proposition

The functor Hom(−,−):𝒜op×𝒜→AbHom(-,-)\colon \mathcal{A}^{op} \times \mathcal{A} \to Ab is a left exact functor, def. 77. In particular for every X∈𝒜X \in \mathcal{A} the functor Hom(X,−):𝒜→AbHom(X,-)\colon \mathcal{A} \to Ab is left exact, and for every A∈𝒜A \in \mathcal{A} the functor Hom(−,A):𝒜op→AbHom(-,A) \colon \mathcal{A}^{op} \to Ab is left exact.

The basic property of the derived Hom-functor/Ext-functor is that it classifies group extensions by (suspensions of) AA. This we now discuss in detail, starting from a basic discussion of group extensions themselves.

The following definition essentially just repeats that of a short exact sequence above in def. 51, but now we consider it for GG a possibly nonabelian group and think of it slightly differently regarding these sequences up to homomorphisms as in def. 86 below. Equivalently we may think of the following as a discussion of the classification of short exact sequences when the leftmost and rightmost component are held fixed.

Proof

Definition

For GG and AAgroups, write Ext(G,A)Ext(G,A) for the set of equivalence classes of extensions of GG by AA, as above and CentrExt(G,A)↪Ext(G,A)CentrExt(G,A) \hookrightarrow Ext(G,A) for for the central extensions. If GG and AA are both abelian, write

AbExt(G,A)↪CentrExt(G,A)
AbExt(G,A) \hookrightarrow CentrExt(G,A)

for the subset of abelian groupsG^\hat G that are (necessarily central) extensions of GG by AA.

We discuss now the following two ways that the Ext1Ext^1 knows about such group extensions.

Central extensions of a possibly non-abelian group GG are classified by the degree-2 group cohomologyHGrp2(G,A)H^2_{Grp}(G,A) of GG with coefficients in AA, and this in turn is equivalently computed by Extℤ[G]Mod1(ℤ,A)Ext^1_{\mathbb{Z}[G] Mod}(\mathbb{Z}, A), where ℤ[G]\mathbb{Z}[G] is the group ring of GG.

Abelian extensions of an abelian gorup GG are classified by ExtAb1(G,A)Ext^1_{Ab}(G,A). In fact, generally, in an abelian category𝒜\mathcal{A} extensions of G∈𝒜G \in \mathcal{A} by A∈𝒜A \in \mathcal{A} (in the sense of short exact sequencesA→G^→G